Second, it must be of constant length — Page 387, Lesson 340

Let’s start with the big picture: we’re measuring time by the Earth’s rotation, and the whole trick is what we measure that rotation against. A ‘day’ is defined as the length of time it takes the Earth to rotate once about its axis, measured against a celestial body — either the Sun or a star. If we measure against a star, we call it ‘sidereal’; if we measure against the Sun, we call it ‘solar’. That’s the fundamental split, and everything else in this lesson hangs off it. Now, a ‘civil’ day — the day we actually live by — has to meet two requirements. First, it must be related to periods of light and darkness, so that 1200 hrs is always about halfway between sunrise and sunset. That means the civil day has to be based on the Sun, not a star. Second, it must be of constant length. So the civil day needs to be solar-based and constant — and as we’ll see, those two things don’t naturally go together. Let’s look at the sidereal day first. A sidereal day is measured against a distant star, and it is of nearly constant length. But here’s the catch: it is not related to light and dark. So although it’s constant, it fails the first requirement — it’s not suitable as a civil day. Now the apparent solar day. This is measured against the real or apparent Sun — the one that actually ‘appears’ to you in the sky. And here’s the problem: using the apparent Sun introduces the issue that the apparent solar day is not a constant length. To understand why, we have to bring in the Earth’s orbit around the Sun. Picture the Earth’s orbit viewed from the North Celestial Pole — the NCP. That’s an imaginary point out in space, located along the continuation of the Earth’s axis, projected from the South pole through the North pole and out into space. So we’re looking down on the orbit from above the North pole. Now imagine the Earth at position A, but in a solar system where the Earth was stationary — no orbit at all. An observer at position Z would have the Sun and a distant star directly over his meridian. After one complete anticlockwise rotation of the Earth, both the Sun and the star would be over his meridian again. In that false, stationary situation, the apparent solar day and the sidereal day would be equal. But that’s a false situation — the Earth doesn’t sit still. In reality, during the period of one 360° revolution, the Earth travels around its orbit to position B. After that 360° rotation, the distant star is again over the observer’s meridian — that’s a sidereal day. But to get the Sun back over the observer’s meridian, the Earth needs an additional rotation and further orbit to position C. So the apparent solar day is longer than the sidereal day. That’s the key relationship: apparent solar day > sidereal day. But remember — the Earth’s orbital speed changes throughout the year. So the amount of that extra rotation needed varies. And that means an apparent solar day cannot be of constant length. It’s solar-based, so it’s related to light and dark, but it fails the constancy requirement. So we need a compromise: the mean solar day. The mean solar day is the average length of an apparent solar day, averaged over the year. It is of constant length, and it is related to light and darkness — so it satisfies both civil-day requirements. It is used as the ‘civil’ day, and it is divided into hours, minutes, and seconds of ‘mean’ time. Here’s a helpful way to think about it: instead of the Earth spinning eastwards, imagine the Sun travelling westwards around the Earth. For mean time, we consider the mean — the average — Sun circling the Earth every 24 hours. That’s the basis of Local Mean Time, or LMT. Now, because the mean Sun is an average, it doesn’t always match the real, apparent Sun. The maximum difference between mean time and apparent Sun time is about 16 minutes, and that occurs in mid-November. A second maximum occurs in mid-February, at about 14 minutes difference. In between those maxima, the difference reduces. And that difference — between mean time and apparent solar time — is known as the Equation of Time. So to tie it all together: we have sidereal days measured against stars, apparent solar days measured against the real Sun, and the mean solar day as the yearly average of the apparent solar day. The mean solar day is our civil day, it’s constant, it’s solar-based, and it gives us Local Mean Time. The Equation of Time is simply the correction between that mean time and the real Sun’s time — peaking at about 16 minutes in mid-November and about 14 minutes in mid-February.